Prime divisors of the Lagarias sequence

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

P

IETER

M

OREE

P

ETER

S

TEVENHAGEN

Prime divisors of the Lagarias sequence

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{1 (2001),}

p. 241-251

<http://www.numdam.org/item?id=JTNB_2001__13_1_241_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Prime divisors of the

_Lagarias

_sequence

par PIETER MOREE et PETER STEVENHAGEN

RÉSUMÉ. Nous donnons une solution à un

problème posé

_par

Lagarias

[5]

en _1985, en déterminant sous GRH la densité de

l’ensemble des nombres _{premiers qui} sont des diviseurs de

ter-mes de la suite définie _par x0 = _{3, x1} = 1 _et la relation

de récurrence _{xn+1 = xn + xn-1.} Cela donne le _premier

exem-ple

d’une suite de récurrence d’ordre 2 _qui n’est pas ’à torsion’ pour

laquelle

on sait déterminer la densité associée des diviseurs

premiers.

ABSTRACT. We solve a 1985

_{challenge problem posed by}

La-garias

[5]

by

determining,

under

_GRH,

the

_density

of the set of

prime numbers that occur as divisor of some term of the _sequence defined

_by

the linear recurrence _xn+1 = xn + xn-1 and

the initial values x0 = 3 and _x1 = 1. This is the first

example

of

a ’non-torsion’ second order _{recurrent sequence} with irreducible

recurrence relation for which we can determine the associated

den-sity of _prime divisors.

1. Introduction

In

_1985,

_Lagarias

_[5]

showed that the set of

_prime

numbers that divide

some Lucas number has a natural

density

2/3

inside the set of all

_prime

numbers. Here the Lucas numbers are the terms of the second order

recur-rent sequence defined

by

the linear recurrence _{= xn} + xn-1

and the initial values xo = 2 and

xl = 1.

Lagarias’s

method is _a

quadratic

analogue

of the

_approach

used

_by

Hasse

_{[2, 3]}

in

_determining,

for a

_given

non-zero

_integer

_a, the

_density

of the set of the

_prime

divisors of the

num-bers of the form an + 1. Note that the _sequence +

_{1 loo 1}

also satisfies

a second order recurrence.

Hasse and

_{Lagarias apply}

the Chebotarev

_density

theorem to a suitable

tower of Kummer fields. Their method of ’Chebotarev

_{partitioning’}

can

be

_adapted

to deal with the class of second order recurrent sequences that

are now known as ’torsion

_sequences’

[1,

_8,

11].

For second order recurrent

integer

sequences that do not

_enjoy

the rather

_special

condition of

_being

’torsion’,

it can no

_longer

be

_applied.

In the case of the the Lucas

_numbers,

changing

the initial values into xo = 3 and

xl = 1

(while

leaving

the

recur-rence _{xn+1 = xn} + xn-1

unchanged)

leads to a _sequence for which

_Lagarias

remarks that his method

_fails,

and he wonders whether some modification

of it can be made to work.

We will

_explain

how non-torsion _sequences lead to a

_question

that is

reminiscent of the Artin

_primitive

root

_conjecture.

In

_particular,

we will see that for a non-torsion sequence, there is no number field F

(of

finite

degree)

with the

_property

that all

_primes

_having

a

_given

splitting

behav-ior in F divide some term of the _sequence. It follows that

Chebptarev

partitioning

can not be

_{applied directly.}

_However,

it is

_possible

to

com-bine the

_technique

of Chebotarev

_partitioning

with the

_{analytic techniques}

employed by Hooley

[4]

in his

_proof

_(under

_assumption

of the

_generalized

Riemann

_hypothesis)

of Artin’s

_primitive

root

_conjecture.

In the case of

the modified Lucas sequence

proposed by Lagarias,

we

_give

a full

analysis

of the situation and _prove the

_following

theorem.

Theorem. Let the

_integer

_sequence

_{defined by}

xo =

3,

_{xl =} 1

and the linear recurrence + xn-l- ·

If

the

generalized

Riemann

hypothesis

holds,

then the set

_of

_prime

numbers that divide some term

of

this _sequence has a natural

_density.

It

_equals

Numerically,

one finds that 45198 out of the first 78498

_primes

below

106

divide the _sequence: a fraction close to .5758.

2. Second order recurrences

Let X = second order _recurrent

sequence. It is our aim to

determine,

whenever it

_exists,

the

_density

_(inside

the set of all

_primes)

of the set of

_prime

numbers _p that divide some term of X.

We let = be the recurrence satisfied

_by

_X,

and denote

by f

=T 2 -

ao E

_Z[T]

the

_{corresponding}

characteristic

_polynomial.

We factor

_f

over an

algebraic

closure of

Q

as

_f

=

(T -

a)(T -

a).

In order to avoid

_{trivialities,}

we will assume that X does not

_satisfy

a

first order recurrence, so that aa =

ao does not vanish. The root

quotient

r =

r( f )

of the _recurrence, which is

only

determined _up to

inversion,

is

then defined as r =

a/a.

It is either a rational number or a

_quadratic

irrationality

of norm 1. In the

separable

case

r ~

1 we have

As our _sequence is

by

assumption

not of order smaller than

_2,

we have

0. Denote

_by

the initial

_{quotient q}

=

q(X)

of X. Just _as the _root

_quotient,

this is _a

num-ber determined _{up to} inversion that is either rational or

quadratic

of norm 1.

The

_elementary

but fundamental observation for second order recurrences

is that for almost all

_primes

_p, we have the fundamental

_equivalence

p divides xn 4=~’

_-c/c

=

(a/a)n

EO/pO

Here 0 is the

_ring

of

_integers

in the field

_{generated by}

the roots of

_{f .}

This is the

_ring

Z if

_f

has rational _roots, and the

_ring

of

_integers

of the

_quadratic

field

_{(a ~X ~ / ( f )}

= otherwise. The

equivalence

above does

not make sense for the

_finitely

_many

_primes

_p for which either r _{or q is} not

invertible modulo _p, but this is irrelevant for

_density

_purposes.

In the

_degenerate

case where the root

_quotient

r is a root of

_unity,

it is

easily

seen that the set of

_primes

_dividing

some term of X is either finite

or cofinite in the set of all

_primes.

We will further exclude this _case, which includes the

_inseparable

case r =

1,

for

which q

is _not defined.

As we are

essentially

interested in the set of

_primes

_p for

_{which q}

is in the

_{subgroup generated by}

r in the finite _group of invertible

residue classes modulo _p, we can formulate the

_problem

we are

_trying

to

solve without _any reference to recurrent _sequences.

_Depending

on whether

the root

_quotient

r is rational or

_quadratic,

this leads to the

_following.

Problem 1. Given two non-zero rational numbers _q and

_{r ~}

_f1,

_compute,

whenever it

_exists,

the

_{density of}

the set

_of

_primes

_p

_for

which we have

q

mod p

E

_(r

_{mod p)}

c

_{FJ§ .}

Problem 2. Let r be a

_quadratic

_{irrationality of}

norm 1 and C~ the

_ring

of

integers

of Q (r).

Given an

edement q

E

_{Q (r)}

_of

norm

_I,

_compute,

whenever

it

_exists,

the

_density

_of

the set

_of

rational

_primes

_p

_for

which we have

q

mod p

E

_(r

_{mod p)}

c

_(O/pO)*.

The instances of the two

_problems

above where

_(q

mod

_r)

is a torsion

ele-ment in the _group

_Q(r)*/(r)

are referred to as torsion cases of the

prob-lem,

and the _sequences that

_give

rise to them are known as torsion

se-quences. The sequences + studied

_by

_Hasse,

the Lucas _sequence +

_{6-nloo o}

with E =

1+2 treated

by Lagarias

and the

Lucas-type

se-quences in

[8]

are

_torsion;

in

_fact,

_they

all

_{have q}

= -1. The main theorem for _{torsion sequences,} for which we refer to

_(11~,

is the

_following.

Theorem. Let X be a second order torsion sequence. Then the set 1rx

of

prime

divisors

of

X has a

positive

rational

density.

3. Non-torsion _sequences

Problem 1 in the

_previous

section is reminiscent of Artin’s famous

_question

on

_primitive

roots:

_given

a non-zero rational number

r ~

~1,

for how

many

primes

p does r

_generate

the _group

_Fp

of units modulo

p?

(One

naturally

excludes the

_finitely

_many

_primes

_p

_dividing

the numerator or

denominator of r from

consideration.)

Artin’s

_conjectural

answer to this

question

is based on the observation that the index

[Fp* :

(r)]

is divisible

by

j

if and

_{only if p splits completely}

in the

_splitting

field

_{Fj =}

_Q((j,

_rl/j)

of the

_polynomial

Xi -

r over

Q. Thus,

r is a

_primitive

root modulo _p if and

only

p does not

split completely

in any of the fields

Fj

with j

&#x3E; 1. For fixed

j,

the set

_Sj

of

_primes

that do

_{split completely}

in

_Fj

has natural

_density

1/[Fj :

Q]

by

the Chebotarev

_density

theorem.

_Applying

an

inclusion-exclusion

_argument

to the sets

_8j,

one

_expects

the set S =

S, B

of

primes

for which r is a

_primitive

root to have natural

_density

J-.

Note that the

_right

hand side of

_(3.1)

_converges for all r E

_{Q* B {±1}}

as

[Fj :

Q]

is a divisor of

cp ( j ) ~ j

with cofactor bounded

by

a constant

depending

only

on r.

A

_{’multiplicative}

version’ of the ’additive formula’

_(3.1)

for

_5(r)

is ob-tained if one starts from the observation that r E

_{Q* B}

is a

_primitive

root if and

_only

_{if P}

does not

_{split completely}

in _any field

_Fe

with i

_prime.

The fields

_Fi

are of

degree

~(~ 2013 1)

for almost all

primes

and

_using

the

fact that

_they

are almost

_{’independent’,}

one can

_successively

eliminate the

primes

that

_{split completely}

in

_{some FE}

to arrive at a heuristic

_density

The correction factor cr for the

’dependency’

between the fields

Ft

is

equal

to 1 if the

_family

of fields is

_linearly

_disjoint

over

Q,

i. _e., if each field

is

_{linearly disjoint}

over

Q

from the

compositum

of the fields

Fe

with

t 0

fOe

If r is not a

_perfect

_{power in}

Q*,

we have =

cr .

It turns out that the

_{only possible}

obstruction to the linear

_disjointness

of the fields

_Ft

occurs when

_F2

= is

quadratic

of odd discriminant.

In this _case,

_F2

is contained in the

_compositum

of the fields

Ft

with t

dividing

its discriminant. The value of cr is a rational

number,

and one can

For

_example,

_taking

r =

5,

_one has

F2

_C

F5

and the

superfluous

’Euler

factor’ 1 -

_{(F5 :}

₁₉

at t = 5 in the

product

IIl(1 -

(Fe :

Ql-’)

is ’removed’

_by

the correction factor c5 =

a 5

=

20

It is non-trivial to make the heuristics above into a

proof.

As

Hooley

[4]

showed,

it can be done if one is

_willing

to assume estimates for the

remainder term in the

_prime

number theorem for the fields

_Fj

that are

currently only

known to hold under

_assumption

of the

_generalized

Riemann

hypothesis.

One should realize that

_only

when we consider

finitely many t

(or j)

at a

_time,

the Chebotarev

density

theorem

_gives

us the densities we

want. After

_taking

a ’limit’ over all

_~,

we

only

know that the

right

hand

side of

_(3.1)

or

(3.2)

is an _upper

density

for the set of

_primes

p for which

r is a

primitive

root. We have however no

_guarantee

that we are left with a

_non-empty

set of such _p. Put somewhat

_differently,

we can not obtain

primes p

for which

_(r

mod

_p)

is a

_primitive

root

_by

_imposing

a

_splitting

condition on _p in a number field F of finite

degree; clearly,

there is

always

some field

_Fe

that is

_{linearly disjoint}

from

_F,

and no

_splitting

condition in

F will

_yield

the ’correct’

_splitting

behavior in

_Ft.

A similar

_phenomenon

occurs in the

analysis

of non-torsion cases of the Problems 1 and 2. This is

exactly

what makes non-torsion sequences so much harder to

_analyze

than torsion sequences.

If

_{(q mod r)}

is not a torsion element in

(a*/(r),

then Problem 1 can

be treated

_by

a

generalization

of the

_arguments

used

by

Artin. For each

integer i &#x3E; 1,

one considers the set of

_primes

_p

_(not

_dividing

the numerator

or denominator of either _q or

r)

for which the index

~FP :

(r)]

is

equal

to

i and the index

_[Fp*

_(q)]

is divisible

_by

i. These are the

primes

that

split

completely

in the field

_Fi~l

=

Q( (i,

rl/i,

ql/i),

but not in any of the fields

Fi,j =

with j

&#x3E; 1. As

_before,

inclusion-exclusion

_yields

a

conjectural

value for the

_density

_8i(r,

_q)

of this set of

_primes,

and

_summing

over i we

_get

as a

_conjectural

value for the

_density

in Problem 1. Note that is

nothing

but the

_primitive

root

_density

_6(r)

from

_(3.1).

The condition that

_(q

mod

_r)

is not a torsion element in

(a*/(r)

means

that q

and r are

_{multiplicatively independent}

in

_Q*.

In this case

Q]

is a divisor of

i2j ~

cp(ij)

with cofactor bounded

_by

a constant

_depending

only

on q and r. Thus the double sum in

(3.3)

_converges, and under GRH one can show

[9, 10]

that its value is indeed the

_density

one is asked to

determine in Problem 1.

As in Artin’s case, one can obtain a

multiplicative

version of

(3.3)

by

a

p)

C

_(r

mod

_p)

in

_Fp

means that for all

_primes

we have an inclusion

(3.4)

(q

mod

_{p) g}

C

_(r

mod

_{p) g}

of the

_~-primary

_parts

of these

_subgroups.

If we fix both t and the number

k =

ordg(p -

1)

of factors t in the order of

Fp,

this condition can be

rephrased

in terms of the

_splitting

behavior of _{p in} the number field

More

_precisely,

we have

ordP(P - 1)

= k if and

only

if _p

splits completely

in

_Q(Cfk)

but not in

_Q((tk+1);

of the

_primes

_p that meet this

_condition,

we want those _p for which the order of the Frobenius elements over _p in

divides the order of the Frobenius elements over

p in

By

the Chebotarev

density

theorem,

one

finds that the set of

_primes

_p with

_{orde(p -}

₁₎

=

k,

which has

density

~-k

for k &#x3E;

_1,

is a union of two sets that each have a

density:

the set of

_primes

p for which the inclusion

(3.4)

holds and the set

of P

for which it does not.

This ’Chebotarev

_{partitioning’}

allows us to

_compute,

for each

_~,

the

_density

of the

_primes

p for which we have the inclusion

(3.4):

_summing

over k in the

previous

argument

yields

a lower

_density,

and this is the

_required

_density

as we can

apply

the same

_argument

to the

_{complementary}

set of

_primes.

For all but

_finitely

_many

_~,

the fields

Q((tk+1,ql/tk)

and

Q((tk+1,rl/tk)

are

_{linearly disjoint}

extensions of with Galois _group for all

k &#x3E; 0. In this case the set of

_primes

_p with

_{ordP(P -}

₁₎

= k

violating

(3.4)

has

_density

Summing

over

k,

we find that

(3.4)

does not hold for a set of

_primes

of

density

~/(~3 -

1).

As the fields

for

_prime

values of £ form a

linearly disjoint family

if we exclude

finitely

many ’bad’

primes t,

the

multiplicative analogue

of

(3.3)

reads

As is shown in

_[9],

_{the ’correction factor’ cq,r} is a rational number that

admits a somewhat involved

description

in terms

_{of q}

and r. In

_practice,

one finds its value most

_{easily by}

_starting

from the additive formula

_{(3.3) .}

In the situation of Problem

_2,

the

_arguments

_{just given}

can be taken over

those rational

_primes

_p that

_split

_completely

in 0.

_Writing

K =

Q(r)

and

we find that the

density

of the rational

primes

_p that are

split

in 0 and for

which we

have q

mod p

E

_(r

_{mod p)}

C

_equals

as in the case of Problem

_1,

one needs to assume the

_validity

of the gen-eralized Riemann

_hypothesis

for this result.

_By

_(3.7),

the

_computation

of

bsplst

amounts to a

_degree

_computation

for the

_family

of fields In

fact,

because of the numerator in

_(3.7)

one _may restrict to the case

where j

is

_squarefree.

As in the case of Problem

_1,

one finds

(under GRH)

that the

_{split density equals}

L

for some rational number

cir.

The next section

_provides

a

typical

_example

of such a

computation.

It shows that the value of

cir

is not as

simple

a

fraction as the

analogous

factor cr

in

(3.2).

For the rational

_primes

_p that are inert in

0,

the determination of the

corresponding density

&#x3E;inert

is more involved than in the

_split

case. The

group in Problem 2 is now

cyclic

of order

p2 -

_1,

and

(q

mod

p)

and

_(r

mod

_p)

are elements of the kernel

of the norm _map, which is

cyclic

of order

_p+1.

In order to have the inclusion

(3.4)

of

_subgroups

_{of Kp for all}

_primes

we fix £ and k =

orde(p

+

_{1) &#x3E;}

1 and

_rephrase

_(3.4)

in terms of the

_splitting

behavior of _p in the

_quadratic

counterpart

B-’-7 1 /-t ,

of

_(3.5).

Let us assume for

_simplicity

that t is an odd

_prime,

and that

K is not the

_quadratic

subfield of

_Q((i).

Then the

_requirement

that _p be inert in K and

_satisfy

_ordt(p

+

₁₎

= k &#x3E; 1 means that the Frobenius

element

_{of p}

in

_{Gal(K((tk+1)/Q)}

is non-trivial on K and has order 2£ when restricted to Let

_Bk

C be the fixed field of the

_subgroup

generated by

such a Frobenius element. Then

Bk

does not contain K or

(a(~e),

and

_Bk

C

_K((tk)

is a

quadratic

extension. Let _ak be the non-trivial

automorphism

of this extension. Then ak acts

by

inversion on and

the norm-1-condition _{on q} and r means that ak also acts

_by

inversion _{on q} and r. The Galois

_equivariancy

of the Kummer

_pairing

shows that the natural action of Qk on is

trivial,

so is abelian over

_Bk.

It is the

_{linearly disjoint compositum}

of

the

_cyclotomic

extension

Bk

C and the abelian extension

For almost

_all

the group is

isomorphic

to

_Z/2fZ

x

Just like in the rational _case, we want those

primes

_p that have

splitting

field

_Bk

inside and for which the order of the Frobenius elements

over _{p in}

Gal(Bk(ql/lk

+ divides the order of the Frobenius

elements over

_{p in}

_By

the Chebotarev

_partition

argument,

we find

again

that for a

’generic’

prime

~,

a fraction

£/(£3 -1)

of

the

_primes

_p that are inert in O violates

(3.4).

Here

_{’generic’}

means that t

is odd and that has

_degree

_{2t3k(t - 1)}

1. Under

_GRH,

one can

again

deduce that the inert

_density

_{6inert (q,}

_r)

_equals

a rational constant

c-

times the infinite Euler

product occurring

in

(3.6)

and

(3.8).

In

_general,

there are various subtleties that need to be taken care of in

the

_analysis

above for £ =

2,

when

2-power

roots

of q

and _{r are}

adjoined

to

K or

K(~2n).

We do not _{go into} them in this _paper. In the

_example

in the

next

section,

we deal with these

complications by combining

a

simple

ad

hoc

_argument

for a few ’bad’ f with the standard treatment for the

_’good’

~.

4. The

_{Lagarias example}

We now treat the

_{explicit example}

of the modified Lucas _sequence which is the

_subject

of the theorem stated in the introduction. The roots of the characteristic

_polynomial

X2 - X -

1 of the recurrence are 6- =

1+2

and

its

_conjugate

6" =

1-2V5.

The initial values _xo = 3 and _xl = 1

yield

an

initial

_{quotient q}

=

1-3E

of the _sequence. As _7rll = 1 - 3e _E 0 =

Z ~e~

has

norm -11,

we find that we have to solve Problem 2 for

We set 1 I as in the

previous

section.

Proof.

As K =

Q( J5)

is the

quadratic

subfield of

(a(~5),

the field has

_degree

_2cp(ij)

over

_Q

if 5 does not

_{divide ij}

and

_degree

_cp(ij)

if it does.

As _q

₌

is a

_quotient

of two non-associate

_prime

elements in 0 and

7rl 1

E is a fundamental unit in

_C,

the

_polynomials

and r are

irreducible in

_K[X]

for all

_{i, j}

E

Z&#x3E;1

by

a standard result as

[6,

Theorem

VI.9. 1].

Moreover,

the extension K C

_generated

_by

a zero of

Xi_q

is

_totally

ramified at the

_primes

of K

_lying

over

11,

whereas the extension

K C

_{generated by}

a zero of

XV -

r is unramified above 11. It

follows that K C

K(qlli,

is of

_degree

_i2j

for all

_{i, j}

E

Z&#x3E;l.

The intersection is contained in the maximal abelian subfield

_Ko

of

_{K (ql/i , rl/ij),}

which

_equals

One

_trivially

_computes

_Ko

n and the lemma follows. 0 We will need the

_preceding

lemma

_only

for

_squarefree

_j.

In this _case, we

simply

have t =

#~d

_E

{5} : dlijl

for odd i.

If we substitute the

_explicit

_degrees

from Lemma 4.1 in

_(3.7),

we find

that the

_split

_density

for our

_{example equals}

where is as in Lemma 4.1. If we set

then the

_expression

above _may be rewritten as

It is

_elementary

to show

_[9,

Theorem

_4.2]

that

_S’m~n

is the rational

_multiple

- A ~ 0

of the universal constant

arithmetic now

_yields

the value

for the

_density

_{(under GRH)}

of the

_primes

p - fl mod 5

dividing

the

78498

_primes

below

106

are

_split

in K and divide our _sequence: a fraction

close to .2601.

For the inert

_primes

of

_K,

which

_satisfy

_{p - ±2} mod

_5,

we have a closer

look at the ’bad’

_{primes 2,}

5 and 11. As for the rational

problem,

we define

the extensions

... , A--...-,

A_--for

_primes

and note

_{that, by}

Lemma

_4.1,

the

_family

_consisting

of the extensions

_Q2Q5Qll

and of I~ is

_{linearly independent}

over K.

Our first observation is that for the inert

_primes

p, the order p + 1 of the _{group Kp} in

_(3.9)

is never divisible

_by

5. Condition

(3.4)

is therefore automatic for the

_prime

=

_5,

and _{we can}

disregard

the

splitting

behavior

of p

in

_SZ5.

We next observe that for inert _p, the element r =

-ê2

satisfies

For

_primes

p - 3 mod

4,

this shows that

(r

mod

p)2

is the

2-Sylow subgroup

of np, so that

(3.4)

is

_again

automatic for £ = 2. When _{we now}

impose

that

the inert

_primes

_congruent

to 3 mod

_4,

which form a set of

primes

of

density

1/4,

have the correct

_splitting

behavior in the extensions

_Of

_{for £ #}

_{2, 5,}

we are

dealing

with a

linearly disjoint family

and find

(under GRH)

that

the set of these

_primes

has

_density

We next consider the inert

_primes

_{p - 1} mod 4. For these _p, the _congruence

(4.3)

shows that

_(r

mod

_p)

has odd order in _rp, so

(3.4)

is satisfied for £ = 2

if and

_only

if the order

_{of q}

=

-7rfi /1 1

in

rp is also odd. As rp is a

_cyclic

group of order p + 1 - 2 mod

_4,

the order

_{of q}

=

(q

mod

p)

is odd if and

only if q

is a _{square in ~p. Let x} E

_O/pO

be a _{square root} of

_q.

If x is in

r,p,

i. e.,

if x has norm 1 in

_Fp,

then its trace x +

_1/x

is in

_Fp,

and we find

that

is a _square modulo _p. If x is not in _{~p, then} x has norm -1 in

_Fp

and

is a _square modulo _p. As 5 is not a _square modulo our inert

_prime

_p, we

deduce

(4.4) (q

mod

_p)

has odd order in _r~P T# - 1 1 is a _square modulo _p.

If _p satisfies the

_equivalent

conditions of

_(4.4),

then _p is a _square modulo

case,

(3.4)

is satisfied for £ =

2, 5

and 11.

Thus,

the _set of inert

primes

p - 1 mod 4

satisfying

the

quadratic

condition

(4.4)

is a set of

_primes

of

_density

_1/8,

and the subset of those _p that have the correct

_splitting

behavior in the extensions

_Qt

_{for £ #}

_{2, 5,11}

has

_{(under GRH)}

_density

Adding

the fractions obtained for the inert

_primes

_congruent

to 3 mod 4 and to 1 mod

_4,

we obtain

Numerically,

one finds that 24781

_primes

out of the 78498

_primes

below

106

are inert in K and divide our sequence: a fraction close to .3157. The sum

_bplit

+

ðinert

is the value

mentioned in the theorem in the introduction.

References

[1] C. BALLOT, Density of prime divisors of linear recurrent sequences. Mem. of the AMS 551,

1995.

[2] H. HASSE, Über die Dichte der Primzahlen p, für die eine _vorgegebene rationale Zahl _{a ~ 0}

von durch eine _{vorgegebene Primzahl l ~} 2 teilbarer bzw. unteilbarer _Ordnung mod p ist.

Math. Ann. 162 _(1965), 74-76.

[3] H. HASSE, Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl

a ~ 0 von _gerader bzw. ungerader Ordnung mod p ist. Math. Ann. 166 (1966), 19-23.

[4] C. HOOLEY, On Artin’s _conjecture. J. Reine Angew. Math. 225 _(1967), 209-220.

[5] J.C. LAGARIAS, The set _{of primes dividing} the Lucas numbers has density 2/3. Pacific J. Math. 118 _(1985), 449-461; Errata Ibid. 162 _(1994), 393-397.

[6] S. LANG, Algebra, 3rd edition. Addison-Wesley, 1993.

[7] H.W. LENSTRA, JR, On Artin’s _conjecture and Euclid’s _{algorithm in global fields.} Inv. Math.

42 _(1977), 201-224.

[8] P. _MOREE, P. STEVENHAGEN, Prime divisors of Lucas sequences. Acta Arith. 82 _(1997),

403-410.

[9] P. MOREE, P. STEVENHAGEN, A two variable Artin _conjecture. J. Number _Theory 85 _(2000),

291-304.

[10] P.J. _STEPHENS, Prime divisors of second order linear recurrences. J. Number _Theory 8

(1976), 313-332.

[11] P. _STEVENHAGEN, Prime densities _for second order torsion sequences. preprint (2000).

Pieter MOREE

Korteweg-de Vries Institute for Mathematics Plantage Muidergracht 24 1018 TV Amsterdam The Netherlands moreeovins - uva - nl Peter STEVENHAGEN Mathematisch Instituut

Universiteit Leiden, P.O. Box 9512 2300 RA Leiden

The Netherlands